I was reading about the classical topological result of Schönflies that Jordan curve in a plane can be extended homeomorphically onto the whole plane. The well-known counterexample of Alexander's horned sphere tells us that the Schönflies' theorem does not extend into higher dimensions, but can we have a weaker version?
Any homeomorphism of the sphere $\mathbb{S^{n-1}}$ onto a simple $(n-1)$-dimensional polygonal surface of $\mathbb{R^n}$ can be extended to a homeomorphism from $\mathbb{R^n}$ onto $\mathbb{R^n}$.
EDIT: J. W. Alexander showed in 1920 that any simple $(n-1)$-dimensional polyhedral (polygonal surface) in $\mathbb{R^3}$ can be extended to a homeomorphism from $\mathbb{R^3}$ onto $\mathbb{R^3}$. The general result was proved in 1958 by B. Mazur in his PhD thesis.
The generalized version of the Schonflies theorem, known as the Brown-Mazur theorem, says that if $S \subset \mathbb R^n$ is homeomorphic to $\mathbb S^{n-1}$, and if there exists a "bicollaring" of $S$, meaning an open neighborhood $U \subset \mathbb R^n$ of $S$ and a homeomorphism $U \approx \mathbb S^{n-1} \times (-1,+1)$ taking $S$ to $S^{n-1} \times \{0\}$, then the conclusion of the Schonflies theorem holds, saying that there exists a homeomorphism from $\mathbb R^n$ to $\mathbb R^n$ taking $\mathbb S^{n-1}$ to $S$.
In your situation, where $S$ is a polygonal hypersurface homeomorphic to $\mathbb S^{n-1}$, I believe it is indeed true that $S$ is bicollared, although I'm a little unsure; if so, then yes, the conclusion of the Schonflies theorem holds.